Lecture 3

Survival analysis

Problem• Do patients survive longer after treatment

A than after treatment B?

• Possible solutions:– ANOVA on mean survival time?– ANOVA on median survival time?

Progressively censored observations

• Current life table– Completed dataset

• Cohort life table– Analysis “on the fly”

First example of the day

Person-year of observation• In total: 15.122 days ~ 41.4y• 11 patients died: 11/41.4y =

0.266 y-1

26.6 death/100y• 1000 patients in 1 y

or • 100 patients in 10y

Mortality rates• 11 of 25 patients died• 11/25 = 44%• When is the analysis done?

1-year survival rate• 6 patients dies the first year• 25 patients started

• 24%

1-year survival rate• 3 patients less than 1 year• 6/(25-3) = 27%• Patient 7• 24% -27%

Actuarial / life table anelysis• Treatment for lung cancer

Actuarial / life table anelysis• A sub-set of 13 patients undergoing the same treatment

Actuarial / life table anelysis• Time interval chosen to

be 3 months

• ni number of patients starting a given period

Actuarial / life table anelysis• di number of terminal

events, in this example; progression/response

• wi number of patients that have not yet been in the study long enough to finish this period

Actuarial / life table anelysis• Number exposed to risk:

ni – wi/2

Assuming that patients withdraw in the middle of the period on average.

Actuarial / life table anelysis• qi = di/(ni – wi/2)

Proportion of patients terminating in the period

Actuarial / life table anelysis• pi = 1 - qi

Proportion of patients surviving

Actuarial / life table anelysis• Si = pi pi-1 ...pi-N

Cumulative proportion of surviving

Conditional probability

Survival curves• How long will a lung

canser patient survive on this particular treatment?

Kaplan-Meier• Simple example with only

2 ”terminal-events”.

Confidence interval of the Kaplan-Meier method

• Fx after 32 months

( ) ii i

i i i

dSE S S

n n d

1

( ) 0.9 0.094910 10 1iSE S

Confidence interval of the Kaplan-Meier method

• Survival plot for all data on treatment 1

• Are there differences between the treatments?

Comparing Two Survival Curves• One could use the confidence

intervals…• But what if the confidence

intervals are not overlapping only at some points?

• Logrank-stats– Hazard ratio

• Mantel-Haenszel methods

Comparing Two Survival Curves• The logrank statistics • Aka Mantel-logrank statistics• Aka Cox-Mantel-logrank statistics

Comparing Two Survival Curves• Five steps to the logrank statistics table

1. Divide the data into intervals (eg. 10 months)

2. Count the number of patients at risk in the groups and in total

3. Count the number of terminal events in the groups and in total

4. Calculate the expected numbers of terminal events e.g. (31-40) 44 in grp1 and 46 in grp2, 4 terminal events.

expected terminal events 4x(44/90) and 4x(46/90)

5. Calculate the total

Comparing Two Survival Curves• Smells like Chi-Square statistics

2

2

all_treatments

O E

E

2 2

2 23 17.07 12 17.934.02

17.07 17.93

1df 0.05p

Comparing Two Survival Curves• Hazard ratio

1 1

2 2

23 17.07Hazard ratio 2.01

12 17.93

O E

O E

Comparing Two Survival Curves• Mantel Haenszel test

• Is the OR significant different from 1?

• Look at cell (1,1)

• Estimated value, E(ai)

• Variance, V(ai)

a b n

ORc d n

row total * column total

grand total

2

( )( )( )( )( )

1i

a c b d a b c dV a

n n

Comparing Two Survival Curves• Mantel Haenszel test

• df = 1; p>0.05

2( )

1.12( )

i i

i

a E aM H

V a

Hazard function

dH

f c

log( )iH S

d is the number of terminal eventsf is the sum of failure timesc is the sum of censured times